Q&A: How is it Possible that People can Learn and Apply University-Level Math Yet Not be Able to Solve Competition Problems?

by Justin Skycak on

Cross-posted from here.

Question

You said this previously: “It’s pretty common for people to learn and apply university-level math yet not be able to solve competition problems.”

Isn’t this sentence a contradiction? Please unpack this sentence?

I agree, weak math graduates can fail to solve competition problems (henceforward CP) at their degree level.

But I am bewildered if B.Sc. grads can’t solve high school (HS) CP, e.g. International Mathematical Olympiad, or if M.Sc. grads can’t solve undergrad CP, e.g. William Lowell Putnam Mathematical Competition.

I am even more bewildered if MSc’s can’t solve HS CP! Or if Ph.D.’s can’t solve undergrad CP! How can it ever be possible that a PhD can’t solve a HS CP problem?

If a graduate can’t solve CP designed for a LOWER degree level, then she failed to learn and apply university-level math. Correct? What’s wrong with my reasoning?

Answer

There’s a huge difference between math degree programs and math competitions.

  • Degrees are about content knowledge. The way a student graduates from a degree program is by learning and evidencing a base level of competency in some further fields of math. The problems here are intentionally selected, organized, scaffolded to be fairly solvable if the student paid attention to what was covered in class.
  • Competitions are about problem-solving insight. It doesn't matter if your base level of competency extends into levels of math further than what's tested in the competition. It's about how good you are with the tools at the level of the competition. The whole point of a competition problem is for it to be very difficult to solve even if you know the underlying content. The goal is to "spread out" students' performance on the basis of their ability to think insightfully about these kinds of tricky problems.

And more concretely: have you seen the caliber of some of these competition problems? I mean, just look at Problem 3 on the 2023 IMO. Sure, if you’re an average student with an undergrad math degree, then you can probably do basic induction and divisibility proofs on the fly, and you’re probably able to (or at least were able to at one point) reproduce proofs of key theorems in real analysis and abstract algebra, but you’re probably going to have no chance against these IMO problems.

And the Putnam?!?! Don’t even get me started on the Putnam. The top scores are generally somewhere around 100 points out of 120 possible, and the median score is usually… wait for it… usually no higher than 2 points out of 120 possible. Typically the median is 1 point, and sometimes it’s literally 0 points. And typically the students who even take the Putnam at all have mathematical ability well above the average math major. If you take a graduating math major who scored a 0 on the Putnam and put them through 2 years of master’s level mathematics, is that going to bring them to the level of collecting half or more (60+) of the total 120 points? Heck no! It would be a “win” to put any</a> points on the board at all.

Addendum. I realized that I haven’t talked explicitly about “application” of math in my answer, and one might argue that being unable to solve competition problems is a symptom of being unable to apply what you’ve learned. The thing is, there are many many ways to apply math, and the vast vast majority have no bearing on competition problems.

Here’s a concrete example. Going back to that average student with an undergrad math degree (who can do basic induction, etc.), suppose that student also took a machine learning class and learned how to fit a mathematical model to data using gradient descent. After graduation they become a data scientist and complete a bunch of work projects in which they fit models to data and use those models to make predictions that are useful to their employer. Clearly, they are applying what they’ve learned – but have they gotten any better at competition math through this process? Nope!